1. Foundations
• Sets
- what is a set: notation, extension property
- subsets, unions, intersections, complements
- Venn diagrams (see also Counting: inclusion-exclusion, below)
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Logic
- informal logic: parsing propositional statements
- formal logic: and, or, not, if-then
- truth tables and deduction
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Operations (see Whole Numbers: Properties of Operations, below)
2. Whole Numbers and their Properties
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Counting
- one to one correspondence
- inclusion-exclusion
- cartesian products
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Factors
- prime numbers
- unique prime factorization
- counting factors
- LCM: how to obtain it; properties of the LCM
- GCD: how to obtain it; properties of the GCD
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Negative Numbers
- interpretation of negative numbers
- rules for manipulation of negative numbers
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Properties of Operations
- abstract operations: commutativity, associativity, identities, inverses
- properties of addition, multiplication, subtraction, division
- consequences of these properties
- distributivity
- mental math consequences
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Place Value
- how our numeration system works
- other bases
- algebraic representation of numbers in terms of their digits
- negative place value and decimals (see Decimals: place value, below)
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Common Algorithms
- justification of addition, subtraction and multiplication algorithms
- understanding alternative algorithms
3. Division and Fractions
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Division
- inverse of multiplication
- repeated subtraction
- division algorithms
- division of fractions (see: Fractions: multiplication and division, below)
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Ratio and Proportion
- proportionality word problems
- recognizing when given data is in terms of proportion
- cross-multiplication and its justification
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Fractions
- models of fractions: number line /measurement, ratio, quotient, operator
- equivalent fractions and cross -multiplication
- mixed versus improper fractions
- adding and subtracting fractions
- multiplication and division: models, procedures
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Decimals
- place value
- terminating decimals to fractions
- classification of fractions as terminating or repeating decimals
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Scientific Notation
- definition and conversion into and out of
- estimatio of physical quantities
- approximate computation in scientific notation
- logarithmic scales : brief intro
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Percentages
- representation as a fraction or decimal
- percentage increase and decrease
4. General Skills and Attitudes
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Problem Solving Strategies
- trial and error
- reducing the problem
- numbers instead of variables
- harnessing physical intuition
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Generalization and Abstraction
- observing patterns
- expressing your observations
- using variables to express and justify observations
- broadening the scope
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Word Problems
- understanding what is asked
- unambiguous usage of variables
- units
- writing equations for a problem
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Broader Context
- NCTM Standards for active learning, justification/proof, verbalizing
- articulate and grammatical mathematical expression
- argumentation
